To determine the domain and range of the function [tex]\( V = f(t) = 25000 - 2700t \)[/tex], let's analyze it step-by-step.
### Domain
The domain of a function refers to all the possible input values (in this case, values of [tex]\( t \)[/tex]) for which the function is defined.
1. Identify the variable: Here, [tex]\( t \)[/tex] represents the number of years.
2. Determine possible values for [tex]\( t \)[/tex]:
- Since [tex]\( t \)[/tex] represents time in years, it has to be non-negative.
- Therefore, [tex]\( t \geq 0 \)[/tex].
Thus, the domain of [tex]\( f(t) \)[/tex] is all non-negative real numbers:
[tex]\[ \text{Domain} = [0, \infty) \][/tex]
### Range
The range of a function is the set of all possible output values (in this case, values of [tex]\( V \)[/tex]).
1. Formulate the relationship: The given function is [tex]\( V = 25000 - 2700t \)[/tex].
2. Find the maximum value of [tex]\( V \)[/tex]:
- When [tex]\( t = 0 \)[/tex], we substitute and get:
[tex]\[ V = 25000 - 2700 \cdot 0 = 25000 \][/tex]
- This is the maximum value because as [tex]\( t \)[/tex] increases, the value of [tex]\( V \)[/tex] will decrease.
3. Find the minimum value of [tex]\( V \)[/tex]:
- As [tex]\( t \)[/tex] increases indefinitely, theoretically, [tex]\( V \)[/tex] would decrease without bound.
- However, in practical terms, the value of a car can't go negative. The minimum practical value of [tex]\( V \)[/tex] would be 0.
- Setting [tex]\( V = 0 \)[/tex], we solve for [tex]\( t \)[/tex]:
[tex]\[ 0 = 25000 - 2700t \][/tex]
[tex]\[ 2700t = 25000 \][/tex]
[tex]\[ t = \frac{25000}{2700} \approx 9.26 \][/tex]
Therefore, the car's value would only be zero or greater, starting from 25000 when new and decreasing to 0 as it becomes older.
Thus, the range of [tex]\( f(t) \)[/tex] is:
[tex]\[ \text{Range} = [0, 25000] \][/tex]
### Conclusion
- Domain: [tex]\([0, \infty)\)[/tex]
- Range: [tex]\([0, 25000]\)[/tex]
